On the other hand, there are over a hundred countries on Earth and thousands of cities, rivers, mountains, etc. Should we narrow the definition of a country or city in order to make geography class less challenging?

You don't mean you've had to learn every single country and their major cities and rivers at school?! Jeeeeeeeeeeeeez! My commiserations! We just had to learn those on our continent, the US (location, capital & great lakes) and the USSR (Location, capital, resources & major climatical zones).